Method for managing crossovers in the tracking of mobile objects, and associated device

ABSTRACT

A method for managing track crossovers and a device for tracking mobile objects that is suitable for implementing the method are provided. The method for managing track crossovers comprises, for each track at a given time, a step Stp1 of testing in order to determine whether the track in question is ambiguous or not at the given time and, if the track is ambiguous, a step Stp2 of specific processing of the estimate of the track.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a National Stage of International patent applicationPCT/EP2015/070210, filed on Sep. 4, 2015, which claims priority toforeign French patent application No. FR 1401991, filed on Sep. 5, 2014,the disclosures of which are incorporated by reference in theirentirety.

FIELD OF THE INVENTION

The present invention relates to the field of the tracking of mobileobjects. The present invention more particularly relates to a method formanaging crossovers in the tracking of mobile objects and to a devicesuitable for implementing such a method. The method and the device mayin particular be applied to track ships and aircraft.

BACKGROUND

Tracking a set of mobile objects with one or more sensors consists inexploiting over time the data output by the group of at least one sensorto construct and update a set of tracks corresponding to the variousobjects perceived by the one or more sensors. It is a question ofgrouping as time passes the measurements made for the various trackedobjects so as to obtain homogeneous groupings, each grouping having tocorrespond to a different object, and each grouping constituting a trackfollowing the same object as time passes. The objects to be tracked andthe one or more sensors may be moving.

The data corresponding to the various mobile objects present in theenvironment monitored by the one or more sensors may for example includeinformation on the position of said objects. Depending on the sensor,the data may be measured values, such as for example an azimuth in thecase of an ESM (electronic support measure) sensor, or measurementvectors such as for example vectors consisting of an azimuth and adistance in the case of a radar sensor, or vectors consisting of anazimuth and an elevation in the case of an optronic sensor. The datavectors may also contain characteristic measurements describing theobjects. In certain cases, the position information may consist of adated position of the sensor and one or more parameters relating to thelocation of the object in space relative to the sensor.

One problem, in the field of the tracking of mobile objects, is that ofcorrectly tracking the progress of the various objects over time. Whensimilar objects, i.e. objects that are indiscriminable by thecharacteristic measurements taken by one or more sensors, move throughspace it is possible that they cross over in terms of their perceptionfrom the carrier of the one or more sensors, i.e. that the measurementsof the relative position of the objects with respect to said carriermomentarily coincide. Ambiguities are then spoken of. In this case, itbecomes difficult, over a more or less long duration, to know whetherthe information produced by the one or more sensors corresponds to oneobject or to another. One problem that arises is that of knowing how togroup and track the measurements when some of them become momentarilyambiguous. Under these conditions, during the processing of the data,mixups may occur and one or more measurements may be attributed toincorrect tracks. This may degrade the quality of the tracking of theobjects and may lead to incorrect characterizations, discontinuities ortrack splitting at the moment of the crossover. In this context, it isadvantageous to have a solution allowing the processing of the trackingto be improved during these crossover situations.

The tracking of objects or of targets is widely treated in theliterature (see for example Yaakov Bar-Shalom, Xiao-Rong Li—“Estimationand Tracking”—Artech House 1993 or Samuel Blackman, RobertPopoli—“Design and Analysis of Modern Tracking Systems”—Artech House1999). This tracking consists in associating newly produced informationwith tracks already produced in the past while taking into account theproximity of the measured values and of measurement noise, and inupdating the tracks.

The tracking of multiple targets (multiple target tracking or MTT) by asensor involves processing that consists in creating or updating trackson the basis of newly acquired data. This is typically done (see inparticular Samuel Blackman, Robert Popoli—“Design and Analysis of ModernTracking Systems”—Artech House 1999) by chaining five functions:processing of new observations, association of the observations withtracks, management of the tracks (initialization, confirmation ordeletion), filtering and prediction (to update the tracks and to enableestimation of positions in the near future), and windowing (in order toallow associations in the near future while restricting the possibleassociations to a volume about the predicted position).

However sometimes a number of tracks may conflict, for example whenthere is an intersection of the volumes about predicted positions. Innearest neighbor (NN) processing, each track is updated by the closestobservation, even if the observation is compatible with a plurality oftracks. In global nearest neighbor (GNN) processing, the association ismade while considering all the associations compatible with thewindowing, but under the constraint that an observation can beassociated only with at most one track.

The kinematic parameters of the tracks are typically updated by Kalmanfiltering (see Yaakov Bar-Shalom, Xiao-Rong Li—“Estimation andTracking”—Artech House 1993) or by interacting multiple model (IMM)processing (see Yaakov Bar-Shalom, Xiao-Rong Li—“Multitarget-multisensortracking: Principles and techniques”—1995 or Samuel Blackman, RobertPopoli—“Design and Analysis of Modern Tracking Systems”—Artech House1999) if it is desired to use an array of Kalman filters in parallel tomake it possible to adapt to a kinematic change. However, S. Blackmannotes that GNN association processing associated with Kalman filteringonly functions well when the targets are widely spaced (see inparticular Samuel Blackman—“Multiple Hypothesis Tracking For MultipleTarget Tracking”—IEEE A&E Systems Magazine, January 2004 vol. 19, no. 1,Part 2: tutorials, p5-18). In situations of conflict between tracks, thecovariance matrix of the Kalman filter may be increased, but this mayfurther increase the conflicts.

It is also known in the prior art to use a joint probabilistic dataassociation (JPDA) approach, as for example described in 1995 by YaakovBar-Shalom et Xiao-Rong Li in “Multitarget-multisensor tracking:Principles and techniques”. This method consists in updating the trackswith all the observations compatible with the windowing using a sum ofthe observations weighted by their probability. The drawback of thismethod is that it tends to make tracks that are closely spacedagglomerate.

Another way of managing cases of conflicts during associations is to usemultiple hypothesis tracking (MHT) techniques such as for exampleintroduced by D. B. Reid in “An algorithm for tracking multipletargets”—IEEE Transactions on Automatic Control, vol. 21, no. 1(February 1976), p 101-104. This type of processing consists, in casesof association ambiguity, in memorizing and maintaining over time allthe combinations of possible successions of associations betweenobservations and tracks. In order to attempt to preserve only thosetracks that persist over time, a score may be defined for thehypotheses, and only the best hypotheses finally adopted. Thesehypothesis scores may be defined by a likelihood ratio or by a loglikelihood ratio of the likelihood that it be a track to the likelihoodthat it be a false alarm. This type of processing generates acombinatorial explosion that may be controlled by clustering, rejectionor grouping of hypotheses, or by retaining only the k best hypotheses.It will be noted that these approaches use a priori probabilityhypotheses on the number of objects or false-alarm rates.

SUMMARY OF THE INVENTION

One aim of the invention is in particular to correct all or some of thedrawbacks of the prior art by providing a solution allowing ambiguitiesrelated to crossovers in the tracking of mobile objects based on manymeasurements generated by one or more sensors to be correctly managed.

To this end, one subject of the invention is a method for managing trackcrossovers implemented by at least one device for tracking mobileobjects comprising a group of at least one sensor, the tracking ofmobile objects consisting in estimating a set of tracks corresponding tothe various mobile objects perceived by the group of at least onesensor, each track corresponding to a set of successive plotsrepresenting the movement of a given mobile object, said methodcomprising, for each track at a given time, a step Stp1 of testing inorder to determine whether the track in question is ambiguous or not atthe given time and, if the track is ambiguous, a step Stp2 of specificprocessing of the estimate of the track.

According to one embodiment the method furthermore comprises a step Stp3of notification of the state of ambiguity of the track under test at thegiven time.

According to one embodiment, a track k is said to be unambiguous at thetime t_(l) if it respects the relationship:

$\begin{matrix}{{\sum\limits_{\underset{j \neq k}{j = 1}}^{K}{{\hat{q}}_{l - 1}^{j}{g\left( {{\hat{C}}_{l - 1}^{k} + {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)} - {\hat{C}}_{l - 1}^{j} - {{\hat{A}}_{l - 1}^{j}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}} < {ɛ \cdot {\hat{r}}_{l - 1}^{k}}} & \;\end{matrix}$where: K is the total number of tracks created up to the time t_(l−1),

{circumflex over (q)}_(l−1) ^(j) is the estimate of the proportion ofplots classed in track j up to the time t_(l−1),

g(.) is a probability density,

Ĉ_(l−1) ^(j)=Â_(l−1) ^(j)t_(l−1)+{circumflex over (B)}_(l−1) ^(j) is theestimate of the position vector of the track j at the time t_(l−1),

Â_(l−1) ^(j) is the estimate of the velocity of the track j at the timet_(l−1) and {circumflex over (B)}_(l−1) ^(j) the estimate of theposition vector of the track j at the time t=0,

ε is a coefficient. In one embodiment this coefficient is 0.1,

{circumflex over (r)}_(l−1) ^(k)={circumflex over (q)}″_(l−1) ^(k)/cwhere {circumflex over (q)}″_(l−1) ^(k) is the estimate of theproportion of the plots X_(n) classed in a track far removed from thetrack k and c the integral of the domain of the admissible measurements.

According to one embodiment, a track k is said to be unambiguous at atime t_(l) if for any track j≠k it respects the relationship:|∥Ĉ _(l−1) ^(k) +Â _(l−1) ^(k)(t _(l) −t _(l−1))−Ĉ _(l−1) ^(j) −Â _(l−1)^(j)(t _(l) −t _(l−1))∥|>Swhere: |∥.∥| represents the norm defined for a vector X=(x_(p)) by

${{X}} = {\sup\left( \frac{x_{p}}{\sigma_{p}} \right)}$where σ_(p) is the standard deviation for the p_(th) component of X inthe density g(X),

S is a predefined threshold.

According to one embodiment, the step of specific processing of theestimates of the tracks in the ambiguous state at a time t_(l) iscarried out by a recursive filter defined by:

$\quad\left\{ \begin{matrix}{{\hat{C}}_{l}^{k} = {{\hat{C}}_{l - 1}^{k} + {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)} + {{\hat{\alpha}}_{l - 1}^{k}M_{1}{g^{\prime}\left( {X_{l} - {\hat{C}}_{l - 1}^{k} - {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}}} \\{{\hat{A}}_{l}^{k} = {{\hat{A}}_{l - 1}^{k} + {{\hat{\alpha}}_{l - 1}^{k}M_{2}{g^{\prime}\left( {X_{l} - {\hat{C}}_{l - 1}^{k} - {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{1}} \right)}} \right)}}}}\end{matrix} \right.$

-   -   where X_(l) is the plot measured at the time t_(l)        -   Ĉ_(l) ^(k)=Â_(l) ^(k)t_(l)+{circumflex over (B)}_(l) ^(k) is            the estimate of the track in question at the time t_(l),        -   {circumflex over (α)}_(l−1) ^(k) is a weighting parameter,        -   M₁ and M₂ are matrix gains ensuring the stability of the            filter,        -   g′(X) is the gradient of the probability density g(.).

According to one embodiment, the parameter {circumflex over (α)}_(l−1)^(k) is defined by

${\hat{\alpha}}_{l - 1}^{k} = \frac{{\hat{r}}_{l - 1}^{k}}{{\hat{r}}_{l - 1}^{k} + {\sum\limits_{\underset{j \neq k}{j = 1}}^{K}{{\hat{q}}_{l - 1}^{j}{g\left( {X_{l} - {\hat{C}}_{l - 1}^{j} - {{\hat{A}}_{l - 1}^{j}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}}}$

-   -   where {circumflex over (r)}_(l−1) ^(k)={circumflex over        (q)}″_(l−1) ^(k)/c.        -   {circumflex over (q)}_(l−1) ^(j) is the proportion of the            plots X_(n) classed in the track j at the time t_(l−1),        -   Ĉ_(l−1) ^(j) is the estimate of the track in question,        -   Â_(l−1) ^(j) is the estimate of the velocity of the track in            question.

According to one embodiment, the parameter {circumflex over (α)}_(l−1)^(k) is equal to 0 when the track is ambiguous and is equal to 1 in thecontrary case.

According to one embodiment, if duration for which a track is consideredto be ambiguous is longer than a predefined threshold, said track isconsidered to be unambiguous.

According to one embodiment, the group of at least one sensor is chosenfrom passive sensors, radar sensors, sonar sensors, optronic sensors orany combination of these sensors.

Another subject of the invention is a device for tracking mobileobjects, which device is suitable for implementing the method asdescribed above, comprising a group of at least one sensor, a trackingmodule that is configured to track over time mobile objects on the basisof plots and to calculate and/or update tracks, a table of the trackscontaining the list of the tracks tracked by the device, a module formanaging crossovers that is configured to apply a set of processingoperations in case of track ambiguity, and a publication moduleconfigured to transmit to the exterior data from the table of thetracks.

According to one embodiment, the group of at least one sensor is chosenfrom passive sensors, radar sensors, sonar sensors, optronic sensors orany combination of these sensors.

BRIEF DESCRIPTION OF THE DRAWINGS

Other particularities and advantages of the present invention willbecome more clearly apparent on reading the following description, whichis given by way of nonlimiting illustration, and with reference to theappended drawings, in which:

FIG. 1 illustrates an exemplary 2-track crossover geometry;

FIG. 2 shows possible steps of the method for managing crossoversaccording to the invention;

FIG. 3 shows an exemplary embodiment of a device for tracking mobileobjects according to the invention.

DETAILED DESCRIPTION

The processing according to the invention may be implemented by a devicefor tracking mobile objects comprising one or more sensors.

The processing for managing crossovers according to the invention isbased on a model characterising a track and the track closest to it fromall the tracks. For this processing, it will be assumed that only twotracks cross and that the other tracks do not interact with these twotracks.

Below, the processing for managing crossovers according to the inventionwill be described for a linear variation in the measurements as afunction of time.

When the measurements output by the one or more sensors are not noisy,each track k describes a straight line of equation A^(k)·t+B^(k) where tdesignates the time, A^(k) designates the vector of the rate of changeof the measurements of track k, and where B^(k) designates the positionvector at the time t=0 of the measurements of track k.

In the presence of noise, for each n, the plot X_(n) at the time t_(n)is a measurement value or a measurement vector that corresponds to oneof K tracks in the presence of an additive white noise of probabilitydensity g(.) that is assumed known. It is therefore possible to write,if the plots X_(n) are considered to vary linearly as a function of timet_(n):X _(n) =A ^(f(n)) t _(n) +B ^(f(n)) +W _(n) ; n=1,2, . . . ,N  (Equation 1)where W_(n) corresponds to the white noise (which is independent of n)of probability density g(.) and f(.) is an unknown function of [1,2, . .. , N] in [1, 2, . . . , K] indicating which track k corresponds to theplot X_(n).

The objective is to find a processing operation that allows thecoefficients A^(k) and B^(k), the index k representing the track, to beestimated by means of the plots X_(n) without knowing either thefunction f(.) or k. Such as stated, this problem leads to acombinatorial explosion and cannot be processed in the form of equation1 because of the many unknowns, in particular the function f(.). It istherefore chosen to simplify this model. To do this, recourse is made toa statistical model taking into account average effects instead of thephysical reality expressed by equation 1.

Two tracks defined by A, B and by A′, B′ are considered to be close inthe vicinity of the time t_(n) if∥(A−A′)t+B−B′∥<threshold for t belonging to the vicinity of t _(n)with

${X}^{2} = {\sum\limits_{p = 1}^{P}\frac{x_{p}^{2}}{\sigma_{p}^{2}}}$where X=(x₁, x₂, . . . , x_(P)), P is the number of components of X andσ_(p) is the standard deviation of x_(p).

For a track close to a crossover situation in the vicinity of the timet_(n), the plot X_(n) belongs either to the track of interest, or to atrack that is close to the track of interest, or to tracks that arefurther removed. It is proposed to write equation 1, and the time t_(n),in the form:X _(n)=ε_(n)(At _(n) +B)+ε′_(n)(A′t _(n) +B′)+ε″_(n) U _(n) +W_(n)  (Equation 2)

-   -   where ε_(n), ε′_(n), ε″_(n) are such that only one of these        three coefficients is nonzero with a respective probability of        these three states q, q′, q″ respecting q+q′+q″=1,        -   A et B correspond to the coefficients of the track of            interest,        -   A′ et B′ correspond to the coefficients of a track that is            close to the track of interest,        -   W_(n) corresponds to the noise, which is white, independent            of n and of probability density g(.),        -   U_(n) models the other further-removed tracks in the form of            a sequence of independent random variables that are equally            distributed over the value domain of the measurements in            order to translate the unknown nature of these tracks and to            formulate the fewest possible hypotheses on their value.

It will be noted that the model of equation 2 is more general than thatof equation 1, because it also takes into account, by the presence ofthe term U_(n), false measurements that correspond to no track.

The probability density of the plots X_(n), i.e. the density modeled byequation 2, is given by the following equation:

$\begin{matrix}{{p\left( X_{n} \right)} = {{{qg}\left( {X_{n} - {At}_{n} - B} \right)} + \left\lbrack {\frac{q^{''}}{c} + {q^{\prime}{g\left( {X_{n} - {A^{\prime}t_{n}} - B^{\prime}} \right)}}} \right\rbrack}} & \left( {{Equation}\mspace{14mu} 3} \right)\end{matrix}$where c is the integral of the domain of the admissible measurements(i.e. of the domain to which the measurements can belong).

$\frac{1}{c}$is the probability density of U_(n).

Because of the independence of the X_(n), it is possible to write:

$\begin{matrix}{{p\left( {X_{1},X_{2},\ldots\mspace{14mu},X_{N}} \right)} = {\prod\limits_{n = 1}^{N}\left( {{{qg}\left( {X_{n} - {At}_{n} - B} \right)} + {\left( {1 - q} \right){h\left( X_{n} \right)}}} \right)}} & \left( {{Equation}\mspace{14mu} 4} \right)\end{matrix}$

with h(.) defined by:

$\begin{matrix}{{h\left( X_{n} \right)} = {\frac{q^{''}}{c} + {q^{\prime}{g\left( {X_{n} - {A^{\prime}t_{n}} - B^{\prime}} \right)}}}} & \left( {{Equation}\mspace{14mu} 5} \right)\end{matrix}$

In equation 4, the number of parameters of the model of equation 1 hasbeen considerably decreased while nonetheless preserving its “nature”.In reality, equation 1 has been incorporated into a family of modelsrepresented by one and the same measure of probability.

In order to estimate the value of the parameters A and B of the tracks,it is proposed to calculate the maximum likelihood estimator of (A, B)from the plots X_(n), with n=1, 2, . . . , N, while assuming q, q′, A′and B′ to be known. This consists in estimating the parameter θ of arandom model characterized by a probability density p(X₁, X₂, . . . ,X_(n), θ) by means of:

$\hat{\theta} = {{Arg}\;\underset{\theta}{Max}\;{p\left( {X_{1},X_{2},\ldots\mspace{14mu},X_{N},\theta} \right)}}$

-   -   where X₁, X₂, . . . , X_(n) are the plots output by the one or        more sensors.

From equations 4 and 5:

${\ln\left( {p\left( {X_{1},X_{2},\ldots\mspace{14mu},X_{N},A,B} \right)} \right)} = {{\sum\limits_{n = 1}^{N}{\ln\left( {1 + {q\;\frac{{g\left( {X_{n} - {At}_{n} - B} \right)} - {h\left( X_{n} \right)}}{h\left( X_{n} \right)}}} \right)}} + {\sum\limits_{n = 1}^{N}{\ln\left( {h\left( X_{n} \right)} \right)}}}$  and${{Arg}\;\underset{A,B}{Max}\;{\ln\left( {p\left( {X_{1},X_{2},\ldots\mspace{14mu},X_{N},A,B} \right)} \right)}} = {{Arg}\;\underset{A,B}{Max}{\sum\limits_{n = 1}^{N}{\ln\left( {1 + {q\;\frac{{g\left( {X_{n} - {At}_{n} - B} \right)} - {h\left( X_{n} \right)}}{h\left( X_{n} \right)}}} \right)}}}$are obtained.

It is proposed to optimize the search for the maximum for the mostdifficult cases, i.e. when only very little of the track of interest canbe seen. This is expressed by values of q close to 0.

After a limited development to the 1st order in q it is found that:

$\begin{matrix}{\left( {\hat{A},\hat{B}} \right) = {{Arg}\;{Max}{\sum\limits_{n = 1}^{N}\frac{g\left( {X_{n} - {At}_{n} - B} \right)}{h\left( X_{n} \right)}}}} & \left( {{Equation}\mspace{14mu} 6} \right)\end{matrix}$

Formula (6) must be understood in the following way: all the localmaxima of the function to be maximized are sought so as to find theparameters (A^(k), B^(k)) of all the tracks. It will also be noted thatthe processing operation of equation 6 assumes the parameters A′, B′, qand q′ contained in the expression of h(.) to be known.

Two cases are worth examining:

A first case corresponds to the situation where, whatever the tracks kand j of 1, 2, . . . , K, and t_(n):

$\begin{matrix}{{\underset{t_{n}}{Min}{{{\left( {A^{k} - A^{j}} \right)t_{n}} + \left( {B^{k} - B^{j}} \right)}}^{2}} > {threshold}} & \left( {{Equation}\mspace{14mu} 7} \right)\end{matrix}$

This case corresponds to the situation where no track crossover occursover the duration of the processing. Then h(X_(n))≈q″/c, which is equalto a constant. The processing operation of equation 6 thereforesimplifies to:

$\begin{matrix}{\left( {\hat{A},\hat{B}} \right) = {{Arg}\;\underset{A,B}{Max}{\sum\limits_{n = 1}^{N}{g\left( {X_{n} - {At}_{n} - B} \right)}}}} & \left( {{Equation}\mspace{14mu} 8} \right)\end{matrix}$

In this first case, no knowledge of the parameters of the other tracksor of q and q′ is required by the processing—the tracks are completelydecoupled.

A second case corresponds to the situation where the condition ofequation 7 is not met. This corresponds to a situation of trackcrossover. As seen above, it is assumed, at the time at which thecrossover occurs, that there are only two tracks that have crossed.

The measurements are divided into subsets or blocks of N measurementsnumbered l=0, 1, 2, . . . so that the plot of index n becomes a plotindexed l N+p where 0≤p<N.

From equations 5 and 6 it is found that for the block l:

$\begin{matrix}{\left( {{\hat{A}}_{l},{\hat{B}}_{l}} \right) = {{Arg}\mspace{11mu}\underset{A,B}{Max}{\sum\limits_{m = 0}^{l}\;{\sum\limits_{p = 0}^{N - 1}\;\frac{g\left( {X_{{mN} + p} - {At}_{{mN} + p} - B} \right)}{\frac{q^{''}}{c} + {q^{\prime}{g\left( {X_{{mN} + p} - {A^{\prime}t_{{mN} + p}} - B^{\prime}} \right)}}}}}}} & \left( {{Equation}\mspace{14mu} 9} \right)\end{matrix}$

So as to define a processing operation that may be applied to all of theblocks, it is assumed that the estimates of rank l−1 necessary tocalculate Â_(l) and {circumflex over (B)}_(l) are known. Therefore,Â_(l−1) ^(k), {circumflex over (B)}_(l−1) ^(k), {circumflex over(q)}_(l−1) ^(k), {circumflex over (q)}″_(l−1) ^(k)/c={circumflex over(r)}_(l−1) ^(k) is known for each track and it is proposed to estimatethe A^(k) and B^(k) of rank l, in light of equation 9, by:

$\begin{matrix}{\left( {{\hat{A}}_{l}^{k},{\hat{B}}_{l}^{k}} \right) = {{Arg}\mspace{11mu}\underset{A,B}{Max}{\sum\limits_{m = 0}^{l}\;{\sum\limits_{p = 0}^{N - 1}\;\frac{g\left( {X_{{mN} + p} - {At}_{{mN} + p} - B} \right)}{\begin{matrix}{{\hat{r}}_{l - 1}^{k} + {\sum\limits_{\underset{j \neq k}{j = 1}}^{K}\;{{\hat{q}}_{l - 1}^{j}{g\left( {X_{{mN} + p} - {{\hat{A}}_{{{({m - 1})}N} + p}^{j}t_{{mN} + p}} -} \right.}}}} \\\left. {\hat{B}}_{{{({m - 1})}N} + p}^{j} \right)\end{matrix}}}}}} & \left( {{Equation}\mspace{14mu} 10} \right)\end{matrix}$

-   -   where c {circumflex over (r)}_(l−1) ^(k) is the proportion of        the plots X_(n) not classed in the track k for n=1,2, . . . ,        (l−1)N, (the proportion of tracks that were ambiguous with the        track k prior to the time of calculation is neglected)        -   {circumflex over (q)}_(l−1) ^(j) is the proportion of plots            X_(n) classed in the track j for n=1,2, . . . , (l−1)N.

For the rank l, the value {circumflex over (q)}_(l) ^(k) is updated forthe track k with the new proportion of plots assigned to the track k.

On account of the fact that it is being considered that only two trackscross, it will also be noted that in the summation, in the denominatorof equation 10, at most one term is nonzero.

The maximization of equation 10 may be obtained in various ways.

In a first way, the maximization is obtained by calculating the functionto be maximized in a grid of values of A and B about the precedingestimate (Â_(l−1) ^(k),{circumflex over (B)}_(l−1) ^(k)).

According to a second method, the maximization is obtained by a gradientalgorithm (Newton method) from (Â_(l−1) ^(k),{circumflex over (B)}_(l−1)^(k)). This method is most often used when the length N of the blocks isequal to 1. In this case the index l and the index n areindistinguishable. C_(l) ^(k)=A^(k)t_(l)+B^(k) is defined, and after thegradient has been taken, the following is found:

$\begin{matrix}\left\{ \begin{matrix}\begin{matrix}{{\hat{C}}_{l}^{k} = {{\hat{C}}_{l - 1}^{k} + {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)} +}} \\{{\hat{\alpha}}_{l - 1}^{k}M_{1}{g^{\prime}\left( {X_{l} - {\hat{C}}_{l - 1}^{k} - {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}\end{matrix} \\{{\hat{A}}_{l}^{k} = {{\hat{A}}_{l - 1}^{k} + {{\hat{\alpha}}_{l - 1}^{k}M_{2}{g^{\prime}\left( {X_{l} - {\hat{C}}_{l - 1}^{k} - {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}}}\end{matrix} \right. & \left( {{Equation}\mspace{14mu} 11} \right)\end{matrix}$

In this written form:

-   -   g′(X) is the gradient of g(.) with respect to X;    -   M₁ and M₂ are two gains (which are possibly matrices if the        matrix of second derivatives is taken into account in the Newton        method) that ensure the stability of the filter defined by        equation 11 and adjust its bandwidth;        -   and {circumflex over (α)}_(l−1) ^(k) is a weighting            parameter dependent on the state of ambiguity of the track            in question, and which is defined by:

$\begin{matrix}{{\hat{\alpha}}_{l - 1}^{k} = \frac{{\hat{r}}_{l - 1}^{k}}{{\hat{r}}_{l - 1}^{k} + {\sum\limits_{\underset{j \neq k}{j = 1}}^{K}\;{{\hat{q}}_{l - 1}^{j}{g\left( {X_{l} - {\hat{C}}_{l - 1}^{j} - {{\hat{A}}_{l - 1}^{j}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}}}} & \left( {{Equation}\mspace{14mu} 12} \right)\end{matrix}$

The influence of this weighting parameter {circumflex over (α)}_(l−1)^(k) may be understood as follows. When there are no track crossovers atthe time t_(l),

${{\sum\limits_{\underset{j \neq k}{j = 1}}^{K}\;{{\hat{q}}_{l - 1}^{j}{g\left( {X_{l} - {\hat{C}}_{l - 1}^{j} - {{\hat{A}}_{l - 1}^{j}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}} = 0},$and {circumflex over (α)}_(l−1) ^(k)=1. The filter of equation 11 thenbehaves just like the conventional filters employed in multi-objecttracking. A copy of the dynamic variation of the tracks is obtained andthe delta (difference between the plot and the estimate of the plot) ispassed through a gate. In this case, the filter of equation 11corresponds to the gradient algorithm for processing equation 8, and themethod proposed here allows the type of gate that must be employed to bespecified.

In case of track crossover, the parameter {circumflex over (α)}_(l−1)^(k) becomes strictly lower than 1 and decreases until the time ofcrossover. This parameter then increases to 1. As the value of thisparameter decreases, the filter takes the plot X less and less intoaccount at the various times. This may be interpreted as a gradualswitchover to “memory” mode, that, for its part, is characterized by{circumflex over (α)}_(l−1) ^(k)=0. This “memorization” is thengradually abandoned as the tracks separate from each other.

FIG. 1 shows, via a graph, the geometry of one exemplary crossover oftwo tracks 11.

In this graph the successive positions 10 of plots forming two tracks 11that cross at a point of intersection 15 are shown. Each plotcorresponds to a measurement vector X delivered by one or more sensors.The successive plots of a track characterize the movement of the trackedmobile object.

On approaching the point of intersection 15, the tracks 11 enter into anarea of ambiguity 12. This area 12 corresponds to the period for whichthe tracks are close, and in which it becomes difficult to know to whichtrack the plots 10 belong.

FIG. 2 shows possible steps of the method for managing crossovers in thetracking of mobile objects according to the invention. This method maycomprise for each track at each time t_(n), a step Stp1 of testing thestate of ambiguity of the track in question in order to determinewhether the track in question is ambiguous or not at the given time and,if the track is ambiguous, a step Stp2 of specific processing of theestimate of the track taking into account the ambiguity of the trackunder test at the given time t_(n).

The method may also comprise a step Stp3 of notification of the state ofambiguity of the track under test at the given time t_(n).

The step Stp1 of detecting states of track ambiguity consists, for eachtrack at the time t_(l), in verifying whether this track is in a stateof ambiguity or not. To do this a track separation criterion is defined.

According to one embodiment, the track k is said to be unambiguous atthe time t_(l) if the following test is passed:

$\begin{matrix}{{\sum\limits_{\underset{j \neq k}{j = 1}}^{K}\;{{\hat{q}}_{l - 1}^{j}{g\left( {{\hat{C}}_{l - 1}^{k} + {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)} - {\hat{C}}_{l - 1}^{j} - {{\hat{A}}_{l - 1}^{j}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}} < {ɛ\mspace{11mu}{\hat{r}}_{l - 1}^{k}}} & \left( {{Equation}\mspace{14mu} 13} \right)\end{matrix}$

-   -   where K is the total number of tracks created up to the time        l−1,    -   Ĉ_(l) ^(j)=Â_(l) ^(j)t_(l)+{circumflex over (B)}_(l) ^(j) is the        estimate of the position vector of the track j at the time        t_(l),    -   Â_(l) ^(j) the estimate of the velocity vector of the track j at        the time t_(l) and {circumflex over (B)}_(l) ^(j) the estimate        of the position vector of the track j at the time t=0.    -   {circumflex over (q)}_(l−1) ^(j) is the estimate of the        proportion of the plots X_(n) classed in the track j for n=1,2,        . . . , (l−1)N,    -   g(.) is the probability density of the measurement noise,    -   ε is a coefficient. In one embodiment this coefficient is 0.1,    -   {circumflex over (r)}_(l−1) ^(k)={circumflex over (q)}″_(l−1)        ^(k)/c where {circumflex over (q)}″_(l−1) ^(k) is the estimate        of the proportion of the plots X_(n) classed in a track far        removed from the track k and c the integral of the domain of the        admissible measurements.

When the track k fails the test, the track k is said to be ambiguous atthe time t_(l).

In this step, an indicator may be updated for the track in question toindicate that, at the given time, the track is in an ambiguous orunambiguous state.

The ambiguity detection is carried out at each time t_(n), the timeinterval between two times t_(n) possibly being regular or not.

According to another embodiment, a track k is said to be unambiguous atthe time t_(l) if for any track j≠k:|∥Ĉ _(l−1) ^(k) +Â _(l−1) ^(k)(t _(l) −t _(l−1))−Ĉ _(l−1) ^(j) −Â _(l−1)^(j)(t _(l) −t _(l−1))∥|>S

-   -   where |∥.∥| the norm defined for a vector X=(x_(p)) by

${{X}} = {\sup\left( \frac{x_{p}}{\sigma_{p}} \right)}$

-   -    where σ_(p) is the standard deviation for the p^(th) component        of X in the density g(X),        -   S is a predefined threshold.

When a track is in a state of ambiguity, a processing operation specificto the management of crossovers is added to the tracking processing.This specific processing operation consists of a recursive processingoperation or recursive filter applied using equation 11, which waspresented above.

According to a first embodiment of the method according to theinvention, the step Stp2 of specific processing of the data relating tothe tracks in a state of ambiguity consists of switching to memory mode,i.e. new measurements are not taken into account when updating a trackduring the period in which this track is in an area of ambiguity. Thisamounts to saying that the parameters of the position-predictingcalculation no longer vary during the state of ambiguity. In otherwords, this amounts to setting the parameter {circumflex over (α)}_(l−1)^(k) of equation 11 to zero when the track is ambiguous in order to nolonger take into account the measurements. When the track is notambiguous, the value of this parameter is set to 1.

According to another embodiment, gradual switchover to “memory” mode isperformed in order to achieve a less abrupt transition with respect tothe preceding method. This softer transition is achieved by calculatingthe parameter {circumflex over (α)}_(l−1) ^(k) using equation 12, whichwas seen above. When the track enters the area of ambiguity, the valueof the parameter {circumflex over (α)}_(l−1) ^(k) begins to graduallydecrease and the plots are taken into account less and less. Likewise,as the tracked object gets further from the point of intersection of thetwo tracks, the value of the parameter {circumflex over (α)}_(l−1) ^(k)will gradually increase to a value substantially equal to 1.

Switching to “memory” mode makes it possible to ensure that the tracksare not deviated by errors due to mixup of the plots between the twotracks at the moment of the crossover.

Advantageously, the implementation of the first method is much lessexpensive in computational terms.

The state of ambiguity is temporary: it starts, lasts and ends dependingon the variation in the measurements measured for the tracked mobileobjects over time. According to one embodiment of the method, when theduration for which a track is considered to be ambiguous exceeds apredefined threshold, this track is considered to be unambiguous inorder to prevent it from remaining in a state of ambiguity for too long.

The two processing variants described above assume a linear variation inthe measurements as a function of time. This provides flexibility forthe tracking of mobile objects. However, in certain cases it may benecessary to take into account an order higher than the second order inorder to be able to better track the movements of these objects—forexample, the acceleration of relative movements could be taken intoaccount.

The processing operations may be generalized, for example, to the thirdorder for plot variations that are locally not entirely linear, and forany order higher than or equal to 2.

Keeping the above notations, equation 1 becomes:X _(n) =D ^(f(n)) t _(n) ² +A ^(f(n)) t _(n) +B ^(f(n)) +W _(n) ; n=1,2,. . . , N  (Equation 14)where the parameter D makes it possible to introduce an accelerationterm that is a function of time.

Equation 9 becomes:

$\begin{matrix}{\left( {{\hat{A}}_{l},{\hat{B}}_{l},{\hat{D}}_{l}} \right) = {{Arg}\mspace{11mu}\underset{A,B,D}{Max}{\sum\limits_{m = 0}^{l}\;{\sum\limits_{p = 0}^{N - 1}\;\frac{g\left( {X_{{mN} + p} - {Dt}_{{mN} + p}^{\; 2} - {At}_{{mN} + p} - B} \right)}{\frac{q^{''}}{c} + {q^{\prime}{g\left( {X_{{mN} + p} - {D^{\prime}t_{{mN} + p}^{\; 2}} - {A^{\prime}t_{{mN} + p}} - B^{\prime}} \right)}}}}}}} & \left( {{Equation}\mspace{14mu} 15} \right)\end{matrix}$

Equation 10 becomes:

$\begin{matrix}{\left( {{\hat{A}}_{l}^{k},{\hat{B}}_{l}^{k},{\hat{D}}_{l}^{k}} \right) = {{Arg}\mspace{11mu}\underset{A,B,D}{Max}{\sum\limits_{m = 0}^{l}\;{\sum\limits_{p = 0}^{N - 1}\;\frac{g\left( {X_{{mN} + p} - {Dt}_{{mN} + p}^{\; 2} - {At}_{{mN} + p} - B} \right)}{\begin{matrix}{{\hat{r}}_{l - 1}^{k} + {\sum\limits_{\underset{j \neq k}{j = 1}}^{K}\;{{\hat{q}}_{l - 1}^{j}{g\left( {X_{{mN} + p} - {{\hat{D}}_{{{({m - 1})}N} + p}^{j}t_{{mN} + p}^{2}} -} \right.}}}} \\\left. {{\hat{A}}_{{{({m - 1})}N} + p}^{j}t_{{mN} + p}{\hat{- B}}_{{{({m - 1})}N} + p}^{j}} \right)\end{matrix}}}}}} & \left( {{Equation}\mspace{14mu} 16} \right)\end{matrix}$

The recursive processing of equation 11 may be similarly generalized, asmay equations 12 and 13.

In a step Stp3, information on the state of track ambiguity is publishedwith the track. According to one embodiment, each track for examplecomprises an indicator indicating the state of ambiguity and thereforewhether the measurement corresponds to an extrapolation or to an actualmeasurement. This makes it possible to know at the output whether thetrack publication is the result of calculations carried out onmeasurements (case for a track outside of an area of ambiguity) or theresult of the memory of the track (case for the track passing through anarea of ambiguity). The downstream processing device to which the dataare published thus knows, among the transmitted data, which areambiguous and which are not.

The method for managing crossovers according to the invention may beapplied to the tracking of mobile objects such as for example aircraftor ships. It may be implemented by one or more mobile object trackingdevices comprising at least one sensor. The one or more sensors may bepassive sensors, radar sensors, sonar sensors, optronic sensors or anycombinations of these sensors.

Another subject of the invention is a device for tracking mobileobjects, said device being able to implement the method for managingcrossovers that was described above.

FIG. 3 shows an exemplary embodiment of a device for tracking mobileobjects according to the invention.

The device comprises at least one sensor 31 that delivers variousmeasurements corresponding to objects moving in the monitored area.According to one embodiment, the one or more sensors 31 may beelectronic support measure (ESM) sensors (capteurs de Mesure deRenseignement Electronique in French), radar sensors, optronic sensorsor any combinations of these sensors in the case of a multisensordevice.

A tracking module 32 takes each measurement vector X_(n) in order toupdate the corresponding tracks in a table 33 of the tracks. This moduletracks mobile objects over time using a table 33 of the trackscontaining information on the tracked objects and using measurementvectors or plots X_(n).

The table 33 of the tracks contains, for each tracked object, actualinformation and information summarizing the measurement vectors receivedrecently. The table 33 of the tracks thus contains the same type ofinformation as the measurements, i.e. one or more pieces of informationon relative position indicating the location of the object in spacerelative to the sensor and optionally pieces of information describingthe tracked objects. The content of the table 33 of the tracks thusvaries over time depending on measurements delivered by the one or moresensors 31.

The device according to the invention also comprises a tracking module32. The aim of this module is firstly to determine to which track, inthe table 33, to attribute a measurement vector, and then to update theinformation of the track using the more recent data contained in themeasurement vector. The tracking module 32 may thus create a new trackif a measurement vector cannot be associated with an existing track inthe table, or possibly delete a track if there have been no updates forthis track for a long time.

A module 34 for managing crossovers takes action on the table of thetracks in case of track ambiguity by applying a set of additionalprocessing operations as seen above. This module 34 complements theobject tracking processing.

The tracking device comprises a publication module 35 configured totransmit to the exterior information on the tracking of the objects. Thepublished data may correspond to any changes, such as for example one ormore new tracks or recent track changes. These data are for exampletransmitted to another system, to one or more processing stages that arefurther downstream (such as a display), to another platform in the casewhere the device is embedded in a multi-platform system, etc.

The various modules described above may be one or more microprocessors,processors, computers or any other suitably programmed equivalent means.

The invention claimed is:
 1. A method for managing track crossoversimplemented by at least one device for tracking mobile objects, themethod comprising: tracking of mobile objects by estimating a set oftracks corresponding to the mobile objects sensed by at least onesensor, each track corresponding to a set of successive plotsrepresenting a movement of a given mobile object, for each track at agiven time, a step of testing with at least one processor in order todetermine whether the track in question is ambiguous or not at the giventime, a track k being said to be unambiguous at a time t_(l) if the atleast one processor determines that it respects the relationship:${\sum\limits_{\underset{j \neq k}{j = 1}}^{K}\;{{\hat{q}}_{l - 1}^{j}{g\left( {{\hat{C}}_{l - 1}^{k} + {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)} - {\hat{C}}_{l - 1}^{j} - {{\hat{A}}_{l - 1}^{j}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}} < {ɛ.\;{\hat{r}}_{l - 1}^{k}}$or if for any track j≠k the at least one processor determines that itrespects the relationship:|∥Ĉ _(l−1) ^(k) +Â _(l−1) ^(k)(t _(l) −t _(l−1))−Ĉ _(l−1) ^(j) −Â _(l−1)^(j)(t _(l) −t _(l−1))∥|>S in which relationships: K is a total numberof tracks created up to a time t_(l−1), {circumflex over (q)}_(l−1) ^(j)is an estimate of a proportion of plots classed in track j up to thetime t_(l−1), g(.) is a probability density, Ĉ_(l−1) ^(j)=Â_(l−1)^(j)t_(l−1)+{circumflex over (B)}_(l−1) ^(j) is an estimate of aposition vector of the track j at the time t_(l−1), Â_(l−1) ^(j) is anestimate of a velocity of the track j at the time t_(l−1) and{circumflex over (B)}_(l−1) ^(j) is an estimate of a position vector ofthe track j at time t=0, ε is a coefficient, {circumflex over (r)}_(l−1)^(k)={circumflex over (q)}_(l−1) ^(″k)/c where {circumflex over(q)}_(l−1) ^(″k) is an estimate of a proportion of plots X_(n) classedin a track far removed from the track k and c is an integral of a domainof admissible measurements |∥.∥| represents a norm defined for a vectorX=(x_(p)) by:${{X}} = {\sup\left( \frac{x_{p}}{\sigma_{p}} \right)}$  whereσ_(p) is a standard deviation for a p^(th) component of X in a densityg(X), S is a predefined threshold,and if the track is determined by theat least one processor to be ambiguous, the at least one processorimplements a step of specific processing of the estimate of the track.2. The method as claimed in claim 1 further comprises a step ofnotification of a state of ambiguity of the track under test at thegiven time.
 3. The method as claimed in claim 1 wherein the step ofspecific processing of the estimates of the tracks in an ambiguous stateat a time t_(l) is carried out by a recursive filter defined by:$\left\{ \begin{matrix}{{\hat{C}}_{l}^{k} = {{\hat{C}}_{l - 1}^{k} + {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)} + {{\hat{\alpha}}_{l - 1}^{k}M_{1}{g^{\prime}\left( {X_{l} - {\hat{C}}_{l - 1}^{k} - {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}}} \\{{\hat{A}}_{l}^{k} = {{\hat{A}}_{l - 1}^{k} + {{\hat{\alpha}}_{l - 1}^{k}M_{2}{g^{\prime}\left( {X_{l} - {\hat{C}}_{l - 1}^{k} - {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}}}\end{matrix} \right.$ where X_(l) is a plot measured at the time t_(l) ,Ĉ_(l) ^(k)=Â_(l) ^(k)t_(l)+{circumflex over (B)}_(l) ^(k) is an estimateof the track in question at the time t_(l), {circumflex over (α)}_(l−1)^(k) is a weighting parameter, M₁ and M₂ are matrix gains ensuring astability of the filter, g′(X) is a gradient of a probability densityg(.).
 4. The method as claimed in claim 3 wherein the parameter{circumflex over (α)}_(l−1) ^(k) is defined by${\hat{\alpha}}_{l - 1}^{k} = \frac{{\hat{r}}_{l - 1}^{k}}{{\hat{r}}_{l - 1}^{k} + {\sum\limits_{\underset{j \neq k}{j = 1}}^{K}\;{{\hat{q}}_{l - 1}^{j}{g\left( {X_{l} - {\hat{C}}_{l - 1}^{j} - {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}}}$where {circumflex over (r)}_(l−1) ^(k)={circumflex over (q)}_(l−1)^(″k)/c. {circumflex over (q)}_(l−1) ^(j) is the proportion of the plotsX_(n) classed in the track j at the time t_(l−1), Ĉ_(l−1) ^(j) is anestimate of the track in question, Â_(l−1) ^(j) is the estimate of thevelocity of the track in question.
 5. The method as claimed in claim 3wherein the parameter {circumflex over (α)}_(l−1) ^(k) is equal to 0when the track is ambiguous and is equal to 1 in a contrary case.
 6. Themethod as claimed in claim 1 wherein if a duration for which a track isconsidered to be ambiguous is longer than a predefined threshold, saidtrack is considered to be unambiguous.
 7. The method as claimed in claim1 wherein the at least one sensor comprises at least one of thefollowing: passive sensors, radar sensors, sonar sensors, optronicsensors, or any combination of these sensors.
 8. A device for trackingmobile objects, the device for tracking mobile objects configured toimplement the method as claimed in claim 1 and comprising: at least onesensor, a tracking processor module that is configured to track overtime mobile objects on a basis of plots and to calculate and/or updatetracks, a table of the tracks containing a list of the tracks tracked bythe device for tracking mobile objects, a processor module for managingcrossovers that is configured to apply a set of processing operations incase of track ambiguity, a track k being said to be unambiguous at atime t_(l) if the processor module determines that it respects therelationship:${\sum\limits_{\underset{j \neq k}{j = 1}}^{K}\;{{\hat{q}}_{l - 1}^{j}{g\left( {{\hat{C}}_{l - 1}^{k} + {{\hat{A}}_{l - 1}^{k}\left( {t_{l} - t_{l - 1}} \right)} - {\hat{C}}_{l - 1}^{j} - {{\hat{A}}_{l - 1}^{j}\left( {t_{l} - t_{l - 1}} \right)}} \right)}}} < {ɛ.\;{\hat{r}}_{l - 1}^{k}}$or if for any track j≠k the processor module determines that it respectsthe relationship:|∥Ĉ _(l−1) ^(k) +Â _(l−1) ^(k)(t _(l) −t _(l−1))−Ĉ _(l−1) ^(j) −Â _(l−1)^(j)(t _(l) −t _(l−1))∥|>S in which relationships: K is a total numberof tracks created up to a time t_(l−1), {circumflex over (q)}_(l−1) ^(j)is an estimate of a proportion of plots classed in track j up to thetime t_(l−1), g(.) is a probability density, Ĉ_(l−1) ^(j)=Â_(l−1)^(j)t_(l−1)+{circumflex over (B)}_(l−1) ^(j) is an estimate of aposition vector of the track j at the time t_(l−1), Â_(l−1) ^(j) is anestimate of a velocity of the track j at the time t_(l−1) and{circumflex over (B)}_(l−1) ^(j) is an estimate of a position vector ofthe track j at time t=0, ε is a coefficient, {circumflex over (r)}_(l−1)^(k)={circumflex over (q)}_(l−1) ^(″k)/c where {circumflex over(q)}_(l−1) ^(″k) is an estimate of a proportion of plots X_(n) classedin a track far removed from the track k and c is an integral of a domainof admissible measurements, |∥.∥| represents a norm defined for a vectorX=(x_(p)) by:${{X}} = {\sup\left( \frac{x_{p}}{\sigma_{p}} \right)}$  whereσ_(p) is a standard deviation for a p^(th) component of X in a densityg(X), S is a predefined threshold, and a publication processor moduleconfigured to externally transmit data from the table of the tracks. 9.The device as claimed in claim 8 wherein the at least one sensorcomprises one of the following: passive sensors, radar sensors, sonarsensors, optronic sensors, or any combination of these sensors.
 10. Thedevice as claimed in claim 8 wherein the publication processor module isconfigured to transmit data from the table of the tracks to anothersystem.
 11. The device as claimed in claim 8 wherein the publicationprocessor module is configured to transmit data from the table of thetracks to another platform.
 12. The device as claimed in claim 8 furthercomprising a display configured to display data from the table of thetracks.
 13. The device as claimed in claim 8 wherein the coefficient εis 0.1.
 14. The method as claimed in claim 1 further comprisingtransmitting with a publication processor module information on thetracking of the objects.
 15. The method as claimed in claim 1 furthercomprising transmitting with a publication processor module informationon the tracking of the objects to another system.
 16. The method asclaimed in claim 1 further comprising transmitting with a publicationprocessor module information on the tracking of the objects to anotherplatform.
 17. The method as claimed in claim 1 further comprisingdisplaying on a display information on the tracking of the objects. 18.The method as claimed in claim 1 wherein the coefficient ε is 0.1.